Qualitative behaviour of numerical methods for SDEs and application to homogenization K. C. Zygalakis Oxford Centre For Collaborative Applied Mathematics, University of Oxford. Center for Nonlinear Analysis, Carnegie Mellon University, 20/10/2011 K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 1 / 45
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Qualitative behaviour of numerical methods for … behaviour of numerical methods for SDEs and application to homogenization K. C. Zygalakis Oxford Centre For Collaborative Applied
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Qualitative behaviour of numerical methods for SDEsand application to homogenization
K. C. Zygalakis
Oxford Centre For Collaborative Applied Mathematics,University of Oxford.
Center for Nonlinear Analysis,Carnegie Mellon University,
20/10/2011
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 1 / 45
Outline
1 Modified Equations
ODE theory.Main idea for SDEs.Different numerical methods and Associated Modified Equations.Numerical examples.
2 Application to Homogenization
Long time behaviour and homogenization.Numerical algorithms/results.From homogenization to averaging in cellular flows.
3 Higher order numerical methods based on modified equations.
Key idea.One simple example.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 2 / 45
Introduction
Motivating example
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 3 / 45
Introduction
Interesting Question
The two numerical methods have the same order of convergence butcompletely different qualitative behaviour.
Is there a way to distinguish between these two methods?
A very powerful tool for addressing this question is backward erroranalysis (modified equations).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 4 / 45
Introduction Ordinary Differential Equations
Modified equations for ODEs
dx
dt= f (x),
and let xn be a numerical approximation of x of order p:
|x(nh)− xn| = O(hp).
Can I find X (t) satisfying another ODE (modified equation) such that:
|X (nh)− xn| = O(hp+q).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 5 / 45
Introduction Ordinary Differential Equations
Euler method-one dimension
xn+1 = xn + hf (xn).
Modified equation:
dX
dt= f (X )− h
2f ′(X )f (X ),
since|X (nh)− xn| = O(h2).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 6 / 45
Introduction Ordinary Differential Equations
Sketch proof
dX
dt= f (X ) + hg(X ).
X (h) = X (0) +
∫ h
0
(f (X (s)) + hg(X (s))) ds
= X (0) + hf (X (0)) + h2g(X (0)) +h2
2f (X (0))f ′(X (0)) +O(h3).
Assume x0 = X (0) then
X (h)− x1 = h2
(g(X (0)) +
1
2f (X (0))f ′(X (0))
)+O(h3),
and thus
g(x) = −1
2f (x)f ′(x).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 7 / 45
Introduction Stochastic Differential Equations
Stochastic Differential Equations and Numerical Methods
dx = u(x)dt + σ(x)dWt ,
Euler method:xn+1 = xn + hu(xn) +
√hσ(xn)ξn,
θ-Milstein method:
xn+1 = xn +θhu(xn+1)+(1−θ)hu(xn)+√
hσ(xn)ξn +h
2σ(xn)σ(1)(xn)(ξ2
n−1),
where ξn ∼ N (0, 1).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 8 / 45
Introduction Stochastic Differential Equations
Weak and Strong Convergence
Weak convergence: We look at |E(φ(x(nh)))− E(φ(xn))|.Strong convergence: We look at E|x(nh)− xn|.In general the weak and strong order of convergence of a numericalmethod NEEDS NOT to be the same!!!
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 9 / 45
Introduction Stochastic Differential Equations
Statement of the Problem
Let x(t) satisfy the following SDE:
dx = u1(x)dt + σ1(x)dWt ,
and xn be its numerical approximation at T = nh by a weak p-ordermethod i.e
|E(φ(x(T )))− E(φ(xn))| = O(hp), ∀φ ∈ C∞.
We want to develop a procedure that allows us to evaluate the propertiesof our weak numerical scheme.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 10 / 45
Introduction Stochastic Differential Equations
Statement of the Problem
Let x(t) satisfy the following SDE:
dx = u1(x)dt + σ1(x)dWt ,
and xn be its numerical approximation at T = nh by a weak p-ordermethod i.e
|E(φ(x(T )))− E(φ(xn))| = O(hp), ∀φ ∈ C∞.
We want to develop a procedure that allows us to evaluate the propertiesof our weak numerical scheme.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 10 / 45
Introduction Stochastic Differential Equations
First Modified Equation
We want to find a modified SDE of the form (i.e., find v2 and σ2)
dx = [u1(x) + hu2(x)] + [σ1(x) + hσ2(x)] dWt ,
for which
|E(φ(x(T )))− E(φ(xn))| = O(hp+1), ∀φ ∈ C∞.
For the rest of the talk we concentrate in the case where p = 1.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 11 / 45
Main Idea
Generators for ODEs and SDEs
ODE:
dx = h(x)dt,
Lu := h(x) · ∇xu.
SDE:
dx = h(x)dt + σ(x)dWt ,
Lu := h(x) · ∇xu +1
2σ(x)σT (x) : ∇x∇xu.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 12 / 45
Main Idea
Backward Kolmogorov Equation
∂u
∂t= Lu,
u(x , 0) = φ(x).
Thenu(x , t) = E(φ(x(t))|x(0) = x).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 13 / 45
Main Idea
Stochastic B-series
By integrating over time the backward Kolmogorov Equation and taking aTaylor expansion of u(x , s) around s = 0, we obtain, (assumingappropriate smoothness of the drift and diffusion term)
u(x , h)− φ(x) =∞∑k=0
hk+1
(k + 1)!Lk+1φ(x).
Note that in the case where φ(x) = x , σ(x) = 0, this expansioncorrespond to the B-series expansion of the ODE
dx = v1(x)dt.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 14 / 45
Main Idea
Local Error/Global Error
A weak first order numerical method has the following expansion
unum(x , h)− φ(x) = hLφ(x) + h2Leφ(x) +O(h3),
and so
u(x , h)− unum(x , h) = h2
(1
2L2φ(x)− Leφ(x)
), Local Error
which implies that
u(x ,T )− unum(x ,T ) = O(h). Global Error
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 15 / 45
Main Idea
Generator of the Modified Equation
Remember that the 1-st modified equation is of the form
dx = [u1(x) + hu2(x)] + [σ1(x) + hσ2(x)] dWt .
Its generator L can be written as
L = L0 + hL1 + h2L2,
where L0 is the generator of the original SDE and
L1φ := u2(x)dφ
dx+ σ1(x)σ2(x)
d2φ
dx2.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 16 / 45
Main Idea
Main Equation
If we now subtract the Taylor expansion of the numerical method from thestochastic B-series of the modified equation we see that in order for thelocal error to be O(∆t3) we need
L1φ = Leφ−1
2L2
0φ, ∀φ ∈ C∞.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 17 / 45
Different Numerical Methods
Euler-Maryama Method
In the case of Euler-Maryama method in the case of multiplicative noise itturns out that a modified equation does not exist since
L1φ 6= · · ·+σ3
1(x)
2σ
(1)1 (x)φ(3)(x).
as L1 is a second order partial differential operator!!!
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 18 / 45
Different Numerical Methods
θ-Milstein Method
u2(x) =
(θ − 1
2
)(v1(x)v
(1)1 (x) +
σ21(x)
2v
(2)1 (x)
),
σ2(x) =
(θ − 1
2
)σ1(x)v
(1)1 (x)− 1
2v1(x)σ
(1)1 (x)− σ2
1(x)
4σ
(2)1 (x).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 19 / 45
Numerical examples SDEs Driven by Multiplicative Noise
Geometric Brownian motion
dx = µxdt + σxdWt ,
dX =
[(µ− h
2µ2
)X
]dt + σX (1− hµ) dWt .
10−3
10−2
10−1
10−5
10−4
10−3
10−2
10−1
100
∆ t
erro
r
original SDEmodified SDE
10−3
10−2
10−1
10−4
10−3
10−2
10−1
100
101
∆ t
erro
r
original SDEmodified SDE
First moment Second moment
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 20 / 45
Numerical examples ∞ Modified equations
Linear SDEs with additive noise
dx = Axdt + ΣdWt ,
Numerical Approximation:
x(h) = A(h)x + f (h, ω).
Example (Euler-Maryama):
A(h) = (I + hA),
f (h, ω) = Σ√
hξ.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 21 / 45
Numerical examples ∞ Modified equations
∞ Modified Equation and its coefficients
dx = Axdt + ΣdWt ,
A =log(A(h))
h,
eAhΣΣT eATh − ΣΣT = AJ + JAT ,
whereJ = E(ff T ).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 22 / 45
Numerical examples ∞ Modified equations
Orstein Uhlenbeck Process
dx = −γxdt + σdWt .
Forward Euler:
A =log(1− γh)
h,
Σ = σ
√2 log(1− γh)
(1− γh)2 − 1.
Backward Euler:
A = − log(1 + γh)
h,
Σ = σ
√2 log(1 + γh)
1− (1 + γh)−2.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 23 / 45
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 39 / 45
Higher order methods
Application to an economy model for asset prices
dX1 = β1X1X2dW1,
dX2 = −(X2 − X3)dt + β2X2dW2,
dX3 = α(X2 − X3)dt,
N. Hofmann, E. Platen, M. Schweizer. Option pricing under incompletness andstochastic volatility. Mathematical Finance, 2(3):153–187, (1992).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 40 / 45
Higher order methods
Numerical Investigations
Error for E(X 21 ). Nonstiff case α = 1. Error for E(X 2
1 ). Very stiff case α = 100.
Error for E(X 21 ). Stiff case α = 25. Error for E(X 2
2 ). Stiff case α = 25.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 41 / 45
Summary
Conclusions
1 It is not always possible to write down a modified Ito SDE for a givennumerical method.
2 In the case of linear SDEs with additive noise it is possible to writedown an ∞-modified equation that the numerical method satisfyexactly in the weak sense.
3 It is possible to generalize ideas from the backward error analysis ofODEs to SDEs.
4 Modified equations can be used as a tool for constructing higher ordermethods.
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 42 / 45
Summary
Future work
1 Find modified equations for numerical methods with respect to strongconvergence.
2 Give a rigorous explanation for failing to find a modified SDE for theEuler method in case of multiplicative noise.
3 Use modified equations to characterize the invariant measureapproximated by different numerical schemes.
4 Compare exit times from a square for different starting points, withthe ones of the effective Brownian motion.
5 Study exit times in case where the inertia is important (inertialparticles).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 43 / 45
Acknowledgements
Thank for your attention!Collaborators:A. Abdulle (EPFL), D. Cohen (Basel), G. Iyer (CMU),G. Pavliotis (Imperial), A. M. Stuart (Warwick), G. Villmart (ENSCachan).
Funding: David Crighton Fellowship and award KUK-C1-013-04, made byKing Abdullah University of Science and Technology (KAUST).
K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 44 / 45
References
Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration, volume 31 of Springer Series in